Pugs can have a few health issues,though not as many as you'd be lead to believe. Most common problem area's are:
The eyes are probably the biggest concern. Due to their size, using a leash attached to a collar can cause lots of strain, so it's recommended to use a harness while walking.
That flat face comes at a cost, making breathing more difficult. They get enough air, they aren't suffocating, but it does make for a lot of snoring, snorting, coughing, weezing, panting, and other crazy noises. It can also be the reason for their relaxed lifestyle, since its hard to run when you can't get enough air. Dust and dirt can irrate this, making it difficult to breath.
As mentioned in the Grooming section, keep those wrinkles clean and you'll have a happy pug. The skin on the face is very sensitive, so using a moist wipe is the best way to keep them clean. They have sensitive skin, so using a mild puppy shampoo is recommened.
And finally, the heat can be fatal for this dog. Because of the short snout and thick double coat, pugs can't get rid of excess heat efficiently, meaning it doesn't have to be baking hot outside to make them uncomfortable. If it's above 80, limit outdoor time to about 15 minutes. Of course, you can take them swimming to help cool them off.
Before you consider getting a pug, ask yourself what you expect from having this kind of dog. Here are a few Characteristics of Pugs: